Finite Group Databasebeta

← #31890874

Subgroups — #31890874

Order 18176.

630 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 630 conjugacy classes of subgroups — too many to draw. Only its 155 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#384×D71C211yesmaximal, characteristic
2#22588317C212yesmaximal
3#22586224C212yesmaximal, complemented
4#22586223C211yesmaximal, characteristic, complemented
5C71×#1296C211yesmaximal, characteristic, complemented
6C2×C4⋊C4×D71—22no—
7C2×C4⋊C4×D71C412yes—
8C2×C42×D71C2211yescharacteristic
9C2×C4⋊(C71⋊C8)C2212yes—
10C2×(C2×D71)⋊C8—22no—
11#17985869C2211yescharacteristic
12#17985868—22no—
13#17985868C412yes—
14#17985753—22no—
15C2×C4⋊C8×C71C2212yes—
16#384×C71C2211yescharacteristic
17C22×C4×D71—22no—
18C22×C4×D71C2×C411yescharacteristic
19C22×C4×D71D412yes—
20C22×C71⋊C8—22no—
21C2×C4⋊Dic71—22no—
22C2×C4⋊Dic71C2×C412yes—
23C2×Dic71⋊C4—22no—
24C2×Dic71⋊C4—24no—
25C4⋊C4×D71—44no—
26C4⋊C4×D71—44no—
27C4⋊C4×D71—44no—
28C2×C4×Dic71C2×C411yescharacteristic
29C2×C4×Dic71D412yes—
30C42×D71—21no—
31C42×D71—22no—
32C42×D71—21no—
33C4⋊(C71⋊C8)—22no—
34C4⋊(C71⋊C8)—22no—
35(C2×D71)⋊C8—44no—
36C22×C8×C71—22no—
37C2×C4⋊C4×C71—22no—
38C2×C4⋊C4×C71C2×C412yes—
39C2×C42×C71C2311yescharacteristic
40C4⋊C8×C71—22no—
41C4⋊C8×C71—22no—
42C23×D71C22⋊C411yescharacteristic
43C22×Dic71—22no—
44C22×Dic71C22⋊C411yescharacteristic
45C22×Dic71C22⋊C412yes—
46C2×C4×D71—22no—
47C2×C4×D71—21no—
48C2×C4×D71—21no—
49C2×C4×D71—22no—
50C2×C4×D71—22no—
51C2×C4×D71—22no—
52C2×C4×D71—22no—
53C2×C4×D71—22no—
54C2×C4×D71—48no—
55C2×C4×D71—44no—
56C2×C4×D71SD1614yes—
57C2×C4×D71Q1614yes—
58C2×C71⋊C8—42no—
59C2×C71⋊C8—42no—
60C2×C71⋊C8—42no—
61C4⋊Dic71—42no—
62C4⋊Dic71—42no—
63C4⋊Dic71—42no—
64Dic71⋊C4—44no—
65Dic71⋊C4—48no—
66C4×Dic71—21no—
67C4×Dic71—21no—
68C4×Dic71—22no—
69C4×Dic71—22no—
70C22×C4×C71—22no—
71C22×C4×C71C22×C411yescharacteristic
72C22×C4×C71C2×D412yes—
73C2×C8×C71—42no—
74C2×C8×C71—42no—
75C2×C8×C71—42no—
76C4⋊C4×C71—42no—
77C4⋊C4×C71—42no—
78C4⋊C4×C71—42no—
79C42×C71#4111yescharacteristic
80C42×C71#4112yes—
81C42×C71#4111yescharacteristic
82C23×C71C2×C22⋊C411yescharacteristic
83C22×D71—21no—
84C22×D71—21no—
85C22×D71—22no—
86C22×D71#10012yes—
87C22×D71#10314yes—
88C2×C4×C71—21no—
89C2×C4×C71—21no—
90C2×C4×C71—42no—
91C2×C4×C71—44no—
92C2×C4×C71#11711yescharacteristic
93C2×C4×C71#11711yescharacteristic
94C2×C4×C71#12312yes—
95C2×C4×C71#12312yes—
96C2×C4×C71C2×SD1612yes—
97C2×C4×C71C2×Q1612yes—
98C2×C4×C71AGL(1,ℤ/8)12yes—
99C2×C4×C71AGL(1,ℤ/8)12yes—
100C2×Dic71—21no—
101C2×Dic71—22no—
102C2×Dic71—21no—
103C2×Dic71—22no—
104C2×Dic71—42no—
105C2×Dic71—44no—
106C2×Dic71#10112yes—
107C2×Dic71#10314yes—
108C4×D71—24no—
109C4×D71—24no—
110C4×D71—24no—
111C4×D71—24no—
112C4×D71—24no—
113C4×D71—24no—
114C4×D71—88no—
115C8×C71—82no—
116C71⋊C8—82no—
117C22×C71#38111yescharacteristic
118C22×C71C2×#10011yescharacteristic
119C22×C71C2×#10111yescharacteristic
120C22×C71C2×#10312yes—
121C22×C71#41812yes—
122C2×D71—24no—
123C2×D71—22no—
124C2×D71—22no—
125C2×D71—24no—
126C2×D71#33314yes—
127C4×C71—21no—
128C4×C71—21no—
129C4×C71—82no—
130C4×C71#48112yes—
131C4×C71#48112yes—
132C4×C71#48412yes—
133C4×C71#48412yes—
134Dic71—22no—
135Dic71—24no—
136Dic71—82no—
137#8725—711nomaximal, Sylow
138C2×C71C2×#33311yescharacteristic
139C2×C71#154912yes—
140C2×C71#155511yescharacteristic
141C2×C71#182111yescharacteristic
142C2×C71#182412yes—
143D71—24no—
144C2×#384—711no—
145#1797—712no—
146#1296—712no—
147#1296—711no—
148#1296D7111yescharacteristic, complemented
149C71#872511yescharacteristic, complemented, Sylow
150C22×C4⋊C4—712no—
151C22×C4⋊C4—1422no—
152C22×C42—711no—
153C2×C4⋊C8—712no—
154C2×C4⋊C8C2×D7112yes—
155C2×C22⋊C8—1422no—
156#384—712no—
157#384—711no—
158#384—1422no—
159#384C2×D7111yescharacteristic
160#336—1422no—
161C23×C4—712no—
162C23×C4—711no—
163C23×C4—1422no—
164C22×C8—22no—
165C22×C8—1422no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—2844no—
168C2×C4⋊C4—2844no—
169C2×C4⋊C4—2844no—
170C2×C4⋊C4—712no—
171C2×C4⋊C4—1422no—
172C2×C4⋊C4—1424no—
173C2×C4⋊C4—1422no—
174C2×C4⋊C4C4×D7112yes—
175C2×C42—711no—
176C2×C42—712no—
177C2×C42—1421no—
178C2×C42—1422no—
179C2×C42—1421no—
180C2×C42C22×D7111yescharacteristic
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—1422no—
184C4⋊C8—1422no—
185C22⋊C8—2844no—
186C24—711no—
187C22×C4—22no—
188C22×C4—2844no—
189C22×C4—2848no—
190C22×C4—711no—
191C22×C4—712no—
192C22×C4—714no—
193C22×C4—714no—
194C22×C4—1422no—
195C22×C4—1422no—
196C22×C4—1422no—
197C22×C4—1421no—
198C22×C4—1421no—
199C22×C4—1422no—
200C22×C4—1422no—
201C22×C4—1422no—
202C22×C4—1422no—
203C22×C4C2×C4×D7111yescharacteristic
204C22×C4D4×D7112yes—
205C2×C8—42no—
206C2×C8—42no—
207C2×C8—42no—
208C2×C8—2842no—
209C2×C8—2842no—
210C2×C8—2842no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—2842no—
215C4⋊C4—2844no—
216C4⋊C4—2848no—
217C4⋊C4—2842no—
218C4⋊C4—2842no—
219C42—1421no—
220C42—1421no—
221C42—1422no—
222C42—1422no—
223C42#1176586411yescharacteristic
224C42#1176586512yes—
225C42#1176586611yescharacteristic
226C23—712no—
227C23—714no—
228C23—1421no—
229C23—1421no—
230C23—1422no—
231C23C22⋊C4×D7111yescharacteristic
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—44no—
236C2×C4—2842no—
237C2×C4—2844no—
238C2×C4—5688no—
239C2×C4—712no—
240C2×C4—714no—
241C2×C4—1424no—
242C2×C4—1424no—
243C2×C4—1421no—
244C2×C4—1422no—
245C2×C4—1421no—
246C2×C4—1422no—
247C2×C4—1424no—
248C2×C4—1424no—
249C2×C4—1424no—
250C2×C4—1424no—
251C2×C4#1455608511yescharacteristic
252C2×C4#1455608611yescharacteristic
253C2×C4#1455611012yes—
254C2×C4#1455611312yes—
255C2×C4SD16×D7112yes—
256C2×C4Q16×D7112yes—
257C2×C4#1455615212yes—
258C2×C4#1455615512yes—
259C8—82no—
260C8—5682no—
261C22—714no—
262C22—1422no—
263C22—1424no—
264C22—1424no—
265C22—1422no—
266C22#1798586111yescharacteristic
267C22#100×D7111yescharacteristic
268C22#101×D7111yescharacteristic
269C22#103×D7112yes—
270C22#1798600212yes—
271C4—21no—
272C4—21no—
273C4—82no—
274C4—5682no—
275C4—1422no—
276C4—1424no—
277C4#1798631612yes—
278C4#1798631912yes—
279C4#1798633312yes—
280C4#1798633612yes—
281C2—1424no—
282C2#333×D7111yescharacteristic
283C2#2258702312yes—
284C2#2258705211yescharacteristic
285C2#2258841911yescharacteristic
286C2#2258842912yes—
subgroupG/H
centerC23C22⋊C4×D71
commutatorC22×C4×C71C22×C4
FrattiniC2×C42C22×D71
FittingC71×#1296C2
socleC23×C71C2×C22⋊C4
prime radicalC23×D71C22⋊C4
cosocleC2×C42×C71C23
hypercenter#1296D71
hypocenterC71#8725
LCSG ▷ C22×C4×C71 ▷ C22×C71 ▷ C71
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1296

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #22586224C2semidirect
4. #22586223C2semidirect
5. C71×#1296C2semidirect
148. #1296D71semidirect
149. C71#8725semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#384×D712
#225883172
#225862242
#225862232
C71×#12962
#872571